1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HIGHLIGHTS OF OPEN CHANNEL HYDRAULICS AND SEDIMENT
TRANSPORT
Dam at Hiram Falls on the Saco River near Hiram, Maine, USA
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SIMPLIFICATION OF CHANNEL CROSS-SECTIONAL SHAPE
floodplain
channel
floodplain
H
B
River channel cross sections have complicated shapes. In a 1D analysis, it is
appropriate to approximate the shape as a rectangle, so that B denotes channel
width and H denotes channel depth (reflecting the cross-sectionally averaged depth
of the actual cross-section). As was seen in Chapter 3, natural channels are
generally wide in the sense that Hbf/Bbf << 1, where the subscript “bf” denotes
“bankfull”. As a result the hydraulic radius Rh is usually approximated accurately by
the average depth. In terms of a rectangular channel,
Rh 
HB
H

H
H
B  2H 
1  2 
B

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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE SHIELDS NUMBER:
A KEY DIMENSIONLESS PARAMETER QUANTIFYING SEDIMENT MOBILITY
b = boundary shear stress at the bed (= bed drag force acting on the flow per unit
bed area) [M/L/T2]
c = Coulomb coefficient of resistance of a granule on a granular bed [1]
Recalling that R = (s/) – 1, the Shields Number * is defined as
b
 
RgD

It can be interpreted as a ratio scaling the ratio impelling force of flow drag acting on
a particle to the Coulomb force resisting motion acting on the same particle, so that

 ~
 bD2
4
D
 c Rg 
3
2
3
The characterization of bed mobility thus requires a quantification of boundary shear
stress at the bed.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
QUANTIFICATION OF BOUNDARY SHEAR STRESS AT THE BED
U = cross-sectionally averaged flow velocity ( depth-averaged
flow velocity in the wide channels studied here) [L/T]
Q
U
BH
u* = shear velocity [L/T]
u 
b

Cf = dimensionless bed resistance coefficient [1]
b
Cf 
U2
Cz = dimensionless Chezy resistance coefficient [1]
Cz 
U
 C f 1/ 2
u
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW
Keulegan (1938) formulation:
Cz 
U
1  H
 Cf 1/ 2  n11 
u
  ks 
where  = 0.4 denotes the dimensionless Karman constant and ks = a roughness
height characterizing the bumpiness of the bed [L].
Manning-Strickler formulation:
Cz 
U
 C f 1/ 2
u
1/ 6
H
  r  
 ks 
where r is a dimensionless constant between 8 and 9. Parker (1991) suggested
a value of r of 8.1 for gravel-bed streams.
Roughness height over a flat bed (no bedforms):
k s  nk Ds90
where Ds90 denotes the surface sediment size such that 90 percent of the
surface material is finer, and nk is a dimensionless number between 1.5 and 3.
For example, Kamphuis (1974) evaluated nk as equal to 2.
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPARISION OF KEULEGAN AND MANNING-STRICKLER RELATIONS
r = 8.1
100
1/ 6
H
Cz  8.1 
 ks 
Cz
Keulegan
10
Parker Version of ManningStrickler
1
1
10
100
H/ks
1000
Note that Cz does not
vary strongly with depth.
It is often approximated
as a constant in broadbrush calculations.
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
TEST OF RESISTANCE RELATION AGAINST MOBILE-BED DATA WITHOUT
BEDFORMS FROM LABORATORY FLUMES
100.00
1/ 6
R 
Cz  8.1 b 
 ks 
Cz
ETH 52
Gilbert 116
10.00
Parker Version of ManningStrickler
1.00
1.00
10.00
Rb/ks
100.00
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NORMAL FLOW
Normal flow is an equilibrium state defined by a perfect balance between the
downstream gravitational impelling force and resistive bed force. The
resulting flow is constant in time and in the downstream, or x direction.
Parameters:
x = downstream coordinate [L]
H = flow depth [L]
U = flow velocity [L/T]
qw = water discharge per unit width [L2T-1]
B = width [L]
Qw = qwB = water discharge [L3/T]
g = acceleration of gravity [L/T2]
 = bed angle [1]
b = bed boundary shear stress [M/L/T2]
S = tan = streamwise bed slope [1]
(cos   1; sin   tan   S)
 = water density [M/L3]
x
bBx
x

H
B
gHxBS
As can be seen from Chapter 3, the
bed slope angle  of the great
majority of alluvial rivers is sufficiently
small to allow the approximations
8
sin   tan   S , cos   1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NORMAL FLOW contd.
Conservation of water mass (= conservation of water volume as water can be
treated as incompressible):
qw  UH
Qw  qwB  UHB
Conservation of downstream momentum:
Impelling force (downstream component of weight of water) = resistive force
gHBx sin   gHBxS  bBx
Reduce to obtain depth-slope
product rule for normal flow:
b  gHS
u  gHS
x
bBx
x

H
B
gHxBS
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ESTIMATED CHEZY RESISTANCE COEFFICIENTS FOR BANKFULL FLOW
BASED ON NORMAL FLOW ASSUMPTION FOR u*
U
Qbf
Czbf   

 u bankf ull Bbf Hbf gHbf S
Hbf
D50
The plot below is from Chapter 3
100
Czbf
, Ĥ 
10
Grav Brit
Grav Alta
Grav Ida
Sand Mult
Sand Sing
1
1
10
100
1000
Ĥ
10000
100000
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELATION BETWEEN qw, S and H AT NORMAL EQUILIBRIUM
Reduce the relation for momentum conservation b = gHS with the resistance form
b = CfU2:
Cf U2  gHS
or
U
g 1/ 2 1/ 2
H S  Cz g H1/ 2S1/ 2
Cf
Generalized Chezy
velocity relation
Further eliminating U with the relation for water mass conservation qw = UH and
solving for flow depth:
1/ 3
 C f q2w
H  
 gS



Relation for Shields stress  at normal equilibrium:
(for sediment mobility calculations)
b
HS
 

RgD RD
 Cf q
  
 g

2
w
1/ 3



S2 / 3
RD
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ESTIMATED SHIELDS NUMBERS FOR BANKFULL FLOW
BASED ON NORMAL FLOW ASSUMPTION FOR b


bf 50
b
Hbf S


, Q̂ 
RgD50 RD50
Qbf
2
gD50 D50
The plot below is from Chapter 3
1.E+01
1.E+00
bf 50
Grav Brit
Grav Alta
Sand Mult
Sand Sing
Grav Ida
1.E-01
1.E-02
1.E-03
1.E+02
1.E+04
1.E+06
1.E+08
Q̂
1.E+10
1.E+12
1.E+14
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RELATIONS AT NORMAL EQUILIBRIUM WITH MANNING-STRICKLER
RESISTANCE FORMULATION
 Cf q
H  
 gS
2
w
U
1/ 6
1/ 3



C f 1/ 2
H
  r  
 ks 
Solve for H
to find
g
g 1/ 2 1/ 2
H S  r 1/ 6 H2 / 3S1/ 2 Solve for U
to find
Cf
ks
 k 1s/ 3 q2w
H   2
  r gS
1 2 / 3 1/ 2
U H S ,
n



3 / 10
g
1
  r 1/ 6
n
ks
Manning-Strickler velocity relation
(n = Manning’s “n”)
Relation for Shields stress  at normal equilibrium:
(for sediment mobility calculations)
k q 


  
  g 
1/ 3 2
s
w
2
r
3 / 10
S7 / 10
RD
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BUT NOT ALL OPEN-CHANNEL FLOWS ARE AT OR CLOSE TO EQUILIBRIUM!
And therefore the calculation of bed shear stress as b = gHS is not always
accurate. In such cases it is necessary to compute the disquilibrium (e.g.
gradually varied) flow and calculate the bed shear stress from the relation
b  CfU2
Flow into standing water (lake or
reservoir) usually takes the form
of an M1 curve.
Flow over a free overfall
(waterfall) usually takes the form
of an M2 curve.
A key dimensionless parameter describing the way
in which open-channel flow can deviate from
normal equilibrium is the Froude number Fr:
U
Fr 
gH
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NON-STEADY, NON-UNIFORM 1D OPEN CHANNEL FLOWS:
St. Venant Shallow Water Equations
x = boundary (bed) attached nearly horizontal coordinate [L]
y = upward normal coordinate [L]
 = bed elevation [L]
S = tan  - /x [1]
H = normal (nearly vertical) flow depth [L]
Here “normal” means “perpendicular to the bed” and has
nothing to do with normal flow in the sense of equilibrium.
Bed and water surface slopes
exaggerated below for clarity.
Relation for water mass conservation
(continuity):
H UH

0
t
x
Relation for momentum conservation:
UH U2H
1 H2


 g
 gH  Cf U2
t
x
2 x
x
H
y
x

15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
HYDRAULIC JUMP
Supercritical (Fr >1) to subcritical (Fr < 1) flow.
Fr  1
Fr  1
flow
supercritical
subcritical
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ILLUSTRATION OF BEDLOAD TRANSPORT
Double-click on the image to see a video clip of bedload transport of 7 mm gravel in a
flume (model river) at St. Anthony Falls Laboratory, University of Minnesota. (Wait a
bit for the channel to fill with water.) Video clip from the experiments of Miguel Wong.
17
rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ILLUSTRATION OF MIXED TRANSPORT OF SUSPENDED LOAD AND BEDLOAD
Double-click on the image to see the transport of sand and pea gravel by a turbidity current
(sediment underflow driven by suspended sediment) in a tank at St. Anthony Falls Laboratory.
Suspended load is dominant, but bedload transport can also be seen. Video clip from
experiments of Alessandro Cantelli and Bin Yu.
18
rte-bookturbcurr.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PARAMETERS CHARACTERIZING SEDIMENT TRANSPORT
qb
qs
qt
=
=
=
qw

s
R
D
*
=
=
=
=
=
=
Volume bedload transport rate per unit width [L2/T]
Volume suspended load transport rate per unit width [L2/T]
qb + qs = volume total bed material transport rate per unit width
[L2/T]
Volume wash load transport rate per unit width [L2/T]
water density [M/L3]
sediment density [M/L3]
(s/) – 1 = sediment submerged specific gravity [1]
characteristic sediment size (e.g. Ds50) [L]
dimensionless Shields number, = (HS)/(RD) for normal flow [1]
Dimensionless Einstein number for bedload transport
qb 
qb
RgD D
Dimensionless Einstein number for total bed material transport
qt 
qt
RgD D
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SOME GENERIC RELATIONS FOR SEDIMENT TRANSPORT
BEDLOAD TRANSPORT RELATIONS (e.g. gravel-bed stream)
Wong’s modified version of the relation of Meyer-Peter and Müller (1948)

b


q  3.97   

 1.5
c
,
c  0.0495
Parker’s (1979) approximation of the Einstein (1950) relation





 1.5
c
qb  11.2( )  1   
 

4.5
,
c  0.03
TOTAL BED MATERIAL LOAD TRANSPORT RELATION (e.g. sand-bed stream)
Engelund-Hansen relation (1967)
qt 
0.05  5 / 2
( )
Cf
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES
Chaudhry, M. H., 1993, Open-Channel Flow, Prentice-Hall, Englewood Cliffs, 483 p.
Crowe, C. T., Elger, D. F. and Robertson, J. A., 2001, Engineering Fluid Mechanics, John Wiley
and sons, New York, 7th Edition, 714 p.
Gilbert, G.K., 1914, Transportation of Debris by Running Water, Professional Paper 86, U.S.
Geological Survey.
Jain, S. C., 2000, Open-Channel Flow, John Wiley and Sons, New York, 344 p.
Kamphuis, J. W., 1974, Determination of sand roughness for fixed beds, Journal of Hydraulic
Research, 12(2): 193-202.
Keulegan, G. H., 1938, Laws of turbulent flow in open channels, National Bureau of Standards
Research Paper RP 1151, USA.
Henderson, F. M., 1966, Open Channel Flow, Macmillan, New York, 522 p.
Meyer-Peter, E., Favre, H. and Einstein, H.A., 1934, Neuere Versuchsresultate über den
Geschiebetrieb, Schweizerische Bauzeitung, E.T.H., 103(13), Zurich, Switzerland.
Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic Research, Stockholm: 39-64.
Parker, G., 1991, Selective sorting and abrasion of river gravel. II: Applications, Journal of
Hydraulic Engineering, 117(2): 150-171.
Vanoni, V.A., 1975, Sedimentation Engineering, ASCE Manuals and Reports on Engineering
Practice No. 54, American Society of Civil Engineers (ASCE), New York.
Wong, M., 2003, Does the bedload equation of Meyer-Peter and Müller fit its own data?,
Proceedings, 30th Congress, International Association of Hydraulic Research, Thessaloniki,
21
J.F.K. Competition Volume: 73-80.